diff --git a/README.html b/README.html index db8d14c..31cca24 100644 --- a/README.html +++ b/README.html @@ -675,14 +675,15 @@ Use these formulas in particular:

Sampling

-

t=nTsTs=1fsxs(t)=x(t)δs(t)=x(t)∑n∈Zδ(t−nTs)=∑n∈Zx(nTs)δ(t−nTs)Xs(f)=X(f)∗∑n∈Zδ(f−nTs)=X(f)∗∑n∈Zδ(f−nfs)B>12fs,2B>fs→Aliasing\begin{align*} +

t=nTsTs=1fsxs(t)=x(t)δs(t)=x(t)∑n∈Zδ(t−nTs)=∑n∈Zx(nTs)δ(t−nTs)Xs(f)=fsX(f)∗∑n∈Zδ(f−nTs)=fsX(f)∗∑n∈Zδ(f−nfs)  ⟹  Xs(f)=∑n∈ZfsX(f−nfs)Sampling (FT)B>12fs  ⟹  2B>fs→Aliasing\begin{align*} t&=nT_s\\ T_s&=\frac{1}{f_s}\\ x_s(t)&=x(t)\delta_s(t)=x(t)\sum_{n\in\mathbb{Z}}\delta(t-nT_s)=\sum_{n\in\mathbb{Z}}x(nT_s)\delta(t-nT_s)\\ - X_s(f)&=X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-\frac{n}{T_s}\right)=X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-n f_s\right)\\ - B&>\frac{1}{2}f_s, 2B>f_s\rightarrow\text{Aliasing}\\ + X_s(f)&=f_s X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-\frac{n}{T_s}\right)=f_s X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-n f_s\right)\\ + \implies X_s(f)&=\sum_{n\in\mathbb{Z}}f_s X\left(f-n f_s\right)\quad\text{Sampling (FT)}\\ + B&>\frac{1}{2}f_s\implies 2B>f_s\rightarrow\text{Aliasing}\\ \end{align*} -

+

Procedure to reconstruct sampled signal

Analog signal x′(t)x'(t) which can be reconstructed from a sampled signal xs(t)x_s(t): Put xs(t)x_s(t) through LPF with maximum frequency of fs/2f_s/2 and minimum frequency of −fs/2-f_s/2. Anything outside of the BPF will be attenuated, therefore nn which results in frequencies outside the BPF will evaluate to 00 and can be ignored.

@@ -708,6 +709,9 @@ Use these formulas in particular:

Nyquist criterion for zero-ISI

Do not transmit more than 2B2B samples per second over a channel of BB bandwidth.

+

Nyquist rate=2BNyquist interval=12B\text{Nyquist rate} = 2B\quad\text{Nyquist interval}=\frac{1}{2B} +

+

By Bob K - Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=94674142

Insert here figure 8.3 from M F Mesiya - Contemporary Communication Systems (Add image to images/sampling.png)

Cannot add directly due to copyright!

diff --git a/README.md b/README.md index 7bcf59e..a019b65 100644 --- a/README.md +++ b/README.md @@ -490,8 +490,9 @@ Use these formulas in particular: t&=nT_s\\ T_s&=\frac{1}{f_s}\\ x_s(t)&=x(t)\delta_s(t)=x(t)\sum_{n\in\mathbb{Z}}\delta(t-nT_s)=\sum_{n\in\mathbb{Z}}x(nT_s)\delta(t-nT_s)\\ - X_s(f)&=X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-\frac{n}{T_s}\right)=X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-n f_s\right)\\ - B&>\frac{1}{2}f_s, 2B>f_s\rightarrow\text{Aliasing}\\ + X_s(f)&=f_s X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-\frac{n}{T_s}\right)=f_s X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-n f_s\right)\\ + \implies X_s(f)&=\sum_{n\in\mathbb{Z}}f_s X\left(f-n f_s\right)\quad\text{Sampling (FT)}\\ + B&>\frac{1}{2}f_s\implies 2B>f_s\rightarrow\text{Aliasing}\\ \end{align*} ``` @@ -542,6 +543,12 @@ Calculate $C_n$ coefficient as follows from $x_p(t)$: Do not transmit more than $2B$ samples per second over a channel of $B$ bandwidth. +```math +\text{Nyquist rate} = 2B\quad\text{Nyquist interval}=\frac{1}{2B} +``` + + + ![By Bob K - Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=94674142](images/Nyquist_frequency_&_rate.svg) ### Insert here figure 8.3 from M F Mesiya - Contemporary Communication Systems (Add image to `images/sampling.png`) diff --git a/README.pdf b/README.pdf index 8a97b3c..ed99660 100644 Binary files a/README.pdf and b/README.pdf differ